This paper investigates the computational limits of modern Hopfield models from a fine-grained complexity perspective. The key contribution is the characterization of a phase transition in the efficiency of all possible modern Hopfield models based on the norm of patterns. Specifically, the authors establish an upper bound criterion for the norm of input query patterns and memory patterns, proving that only below this criterion can sub-quadratic (efficient) variants of modern Hopfield models exist, assuming the Strong Exponential Time Hypothesis (SETH). To demonstrate their theory, the authors provide an efficient construction of modern Hopfield models using low-rank approximation when the efficient criterion holds. This includes deriving a lower bound on the computational time, which scales linearly with the maximum number of stored memory patterns and the length of the input query sequence. Additionally, they prove the memory retrieval error bound and show that the model achieves almost-linear-time efficiency while maintaining the exponential memory capacity characteristic of modern Hopfield models. The paper also discusses the practical significance of these findings in the context of transformer attention mechanisms and large foundation models.This paper investigates the computational limits of modern Hopfield models from a fine-grained complexity perspective. The key contribution is the characterization of a phase transition in the efficiency of all possible modern Hopfield models based on the norm of patterns. Specifically, the authors establish an upper bound criterion for the norm of input query patterns and memory patterns, proving that only below this criterion can sub-quadratic (efficient) variants of modern Hopfield models exist, assuming the Strong Exponential Time Hypothesis (SETH). To demonstrate their theory, the authors provide an efficient construction of modern Hopfield models using low-rank approximation when the efficient criterion holds. This includes deriving a lower bound on the computational time, which scales linearly with the maximum number of stored memory patterns and the length of the input query sequence. Additionally, they prove the memory retrieval error bound and show that the model achieves almost-linear-time efficiency while maintaining the exponential memory capacity characteristic of modern Hopfield models. The paper also discusses the practical significance of these findings in the context of transformer attention mechanisms and large foundation models.